416 THE THEORY OF TERRESTRIAL MAGNETISM. 



Differentiating m times, we get 



_ 



^r+n-l nn-l p 



(2n- 1) (2w + 3) ^ (2w- 1) (2n + 1) i 



f ! ! n 



D P - - DP 



2n +1 \2n+3 n +* ~ 2n + -S "J 2 ^ 7 ^" 



Differentiating m 1 times, we get 



l - 



Also p^ 



Differentiating we get 



DP - 1 D-P TPP 



Ln -^n+\ L ^ n+I 2^+1^ 



p - _ L_ jy-p D-P 



n * 



2n + 3^ "J 



J: I ] D *P - J- 

 2w+l \2n-l B 2n-l 



D-P 4- 



(2w+l)(2w + 3) ' (2n-l)(2n+8) 



Differentiate this equation m 2 times, then multiply the result by 

 m(m-l) and subtract from 2m[/u,Z)" l ~ 1 P,, + (m- 1) Z) m ' 2 P n ] and we get 



2mp.If- 1 P n + m (m - 1) D'" - a P B 



_ 2m (n + l)-m(m-l) , 

 (2n + l)(2n+3) 



. 2w + 2m (m- 1) _ 



r (27i-l)(2w + 3) " 



m(2n + 3)-m 2 2m 2 m(2n-l) + m !i n ,,, p 



l)(2n + 3) T (2w- 1) (2n + 3) n 



